<head>
<title>Transverse Mercator</title>
</head>
<body>

<h1>Transverse Mercator</h1>

<table border>

<td>Name
<td>Transverse Mercator
<tr>

<td>
<td>Gauss-Kruger
<tr>

<td>EPSG Code
<td>9807
<tr>

<td>GeoTIFF Code
<td>CT_TransverseMercator (1)
<tr>

<td>OGC WKT Name
<td>Transverse_Mercator
<tr>

<td>Supported By
<td>EPSG, GeoTIFF, PROJ.4, OGC WKT
<tr>

</table>

<h3>Projection Parameters</h3>

<table border>
<th>Name
<th>EPSG #
<th>GeoTIFF ID
<th>OGC WKT
<th>Units
<th>Notes

<tr>
<td>Latitude of natural origin
<td>1
<td>NatOriginLat
<td>latitude_of_origin
<td>Angular
<td>

<tr>
<td>Longitude of natural origin
<td>2
<td>NatOriginLong
<td>central_meridian
<td>Angular
<td>

<tr>
<td>Scale factor at natural origin
<td>5
<td>ScaleAtNatOrigin
<td>scale_factor
<td>Unitless
<td>

<tr>
<td>False Easting
<td>6
<td>FalseEasting
<td>false_easting
<td>Linear
<td>

<tr>
<td>False Northing
<td>7
<td>FalseNorthing
<td>false_northing
<td>Linear
<td>

</table>

<h3>PROJ.4 Organization</h3>

<b>
<pre>
  +proj=tmerc +lat_0=<i>Latitude of natural origin</i> 
              +lon_0=<i>Longitude of natural origin</i>
              +k=<i>Scale factor at natural origin</i> 
              +x_0=<i>False Easting</i>
              +y_0=<i>False Northing</i>
</pre>
</b>

<h3>EPSG Notes</h3>

The Transverse Mercator projection in its various forms is the most widely used projected 
coordinate system for world topographical and offshore mapping. All versions have the 
same basic characteristics and formulas. The differences which distinguish the different 
forms of the projection which are applied in different countries arise from variations in the 
choice of the coordinate transformation parameters, namely the latitude of the origin, the 
longitude of the origin (central meridian), the scale factor at the origin (on the central 
meridian), and the values of False Easting and False Northing, which embody the units of 
measurement, given to the origin. Additionally there are variations in the width of the 
longitudinal zones for the projections used in different territories. 

The following table indicates the variations in the projection parameters which distinguish 
the different forms of the Transverse Mercator projection and are used in the Epicentre 
Transverse Mercator projection method:
<p>

<table border>
<th>Name
<th>Areas used
<th>Central meridian(s)
<th>Latitude of origin
<th>CM Scale Factor
<th>Zone width
<th>False Easting at origin
<th>False Northing at origin

<tr>
<td>Transverse Mercator
<td>Various, world wide
<td>Various
<td>Various
<td>Various
<td>Usually less than 6*
<td>Various
<td>Various

<tr>
<td>Transverse Mercator south oriented
<td>South Africa
<td>2* intervals E of 11*E
<td>0*
<td>1.000000
<td>2*
<td>0m
<td>0m

<tr>
<td>UTM North hemisphere
<td>World wide
<td>6* intervals* E & W of 3* E & W
<td>Always 0*	
<td>Always 0.9996
<td>Always 6*
<td>500000m
<td>0m

<tr>
<td>UTM South hemisphere
<td>World wide	
<td>6* intervals E & W of 3* E  & W	
<td>Always 0*	
<td>Always 0.9996
<td>Always 6*
<td>500000m
<td>10000000m

<tr>
<td>Gauss-Kruger
<td>Former USSR Yugoslavia, Germany, S. America	
<td>Various, according to area of cover
<td>Usually 0*
<td>Usually 1.000000
<td>Usually less than 6*, often less than 4*	
<td>Various but often 500000 prefixed by zone number	
<td>Various

<tr>
<td>Gauss Boaga
<td>Italy
<td>Various
<td>Various
<td>0.9996
<td>6*
<td>Various	
<td>0m

</table>
<p>

The most familiar and commonly used Transverse Mercator is the Universal Transverse 
Mercator (UTM) whose natural origin lies on the equator. However, some territories use 
a Transverse Mercator with a natural origin at a latitude closer to that territory.  

In Epicentre the coordinate transformation method is the same for all forms of the 
Transverse Mercator projection. The formulas to derive the projected Easting and 
Northing coordinates are in the form of a series as follows:
<p>

<pre>

	Easting, E =  FE + k0*[A + (1 - T + C)A3/6 + (5 - 18T + T2 + 72C - 58e'2)A5/120]	

	Northing, N =  FN + k0{M - M0 + *tan*[A2/2 + (5 - T + 9C + 4C2)A4/24 + 
				(61 - 58T + T2 + 600C - 330e'2)A6/720]} 
where	T = tan2*
	C = e2 cos2*/(1 - e2) = e'2 cos2*
	A = (* - *0)cos*, with * and *0 in radians
	M = a[(1 - e2/4 - 3e4/64 - 5e6/256 -....)* - (3e2/8 + 3e4/32 + 45e6/1024+....)sin2* 
		+ (15e4/256 + 45e6/1024 +.....)sin4* - (35e6/3072 + ....)sin6* + .....]
	with * in radians and M0 for *0, the latitude of the origin, derived in the same way.
</pre>

The reverse formulas to convert Easting and Northing projected coordinates to latitude 
and longitude are:
<p>

<pre>
	
	* = *1 - (*1tan*1/*1)[D2/2 - (5 + 3T1 + 10C1 - 4C12 - 9e'2)D4/24
			+ (61 + 90T1 + 298C1 + 45T12 - 252e'2 - 3C12)D6/720]
	* = *0 + [D - (1 + 2T1 + C1)D3/6 + (5 - 2C1 + 28T1 - 3C12 + 8e'2  			
			+ 24T12)D5/120] / cos*1
where *1 may be found as for the Cassini projection from:
		
	*1 = *1 + (3e1/2 - 27e13/32 +.....)sin2*1 + (21e12/16 -55e14/32 + ....)sin4*1
		+ (151e13/96 +.....)sin6*1 + (1097e14/512 - ....)sin8*1 + ......
and where
	e1 = [1- (1 - e2)1/2]/[1 + (1 - e2)1/2]
	*1 = M1/[a(1 - e2/4 - 3e4/64 - 5e6/256 - ....)]
	M1 = M0 + (N - FN)/k0
	T1 = tan2*1
	C1 = e'2cos2*1
	D = (E - FE)/(*1k0), with *1 = * for *1
</pre>

For areas south of the equator the value of latitude * will be negative and the formulas 
above, to compute the E and N,  will automatically result in the correct values. Note that 
the false northings of the origin, if the equator, will need to be large to avoid negative 
northings and for the UTM projection is in fact 10,000,000m. Alternatively, as in the case 
of Argentina's Transverse Mercator (Gauss-Kruger) zones, the origin is at the south pole 
with a northings of zero. However each zone central meridian takes a false easting of 
500000m prefixed by an identifying zone number. This ensures that instead of points in 
different zones having the same eastings, every point in the country, irrespective of its 
projection zone, will have a unique set of projected system coordinates. Strict application 
of the above formulas, with south latitudes negative, will result in the derivation of the 
correct Eastings and Northings. 

Similarly, in applying the reverse formulas to determine a latitude south of the equator, a 
negative sign for * results from a negative *1 which in turn results from a negative M1.","For Projected Coordinate System OSGB 1936 / British National Grid
<p>
<pre>
Parameters:
Ellipsoid  Airy 1830  a = 6377563.396 m  1/f = 299.32496
then e'^2 = 0.00671534 and e^2 = 0.00667054

Latitude Natural Origin         49o00'00""N   = 0.85521133 rad
Longitude Natural Origin        2o00'00""W  = -0.03490659 rad
Scale factor ko                     0.9996013                                                                                              False Eastings FE                 400000.00 m
False Northings FN              -100000.00 m

Forward calculation for: 
Latitude       50o30'00.00""N  = 0.88139127 rad
Longitude    00o30'00.00""E  = 0.00872665 rad
A  = 0.02775415       C = 0.00271699
T =  1.47160434       M = 5596050.46
M0 = 5429228.60     nu  = 6390266.03

Then Easting E =        577274.99 m
          Northing N =       69740.50 m

Reverse calculations for same easting and northing first gives :
e1 =    0.00167322      mu1 = 0.87939562
M1 = 5599036.80        nu1 = 6390275.88
phi1  = 0.88185987      D = 0.02775243
rho1 =6372980.21       C1 =  0.00271391
T1 = 1.47441726

Then Latitude       = 50o30'00.000""N
         Longitude    = 00o30'00.000""E"
</pre>

</body>
